2Inventory DefinedInventory is the stock of any item or resource held to meet future demand and can include: raw materials, finished products, component parts, supplies, and work-in-process3
3Inventory Classifications Process stageDemand TypeNumber & ValueOtherRaw MaterialWIPFinished GoodsIndependent DependentA Items B Items C ItemsMaintenance Operating
4Independent vs. Dependent Demand Independent Demand (Demand for the final end-product or demand not related to other items; demand created by external customers)FinishedproductIndependent demand is uncertain Dependent demand is certainADependent Demand(Derived demand for component parts,subassemblies,raw materials, etc- used to produce final products)B(4)C(2)D(1)E(1)E(2)B(1)E(3)Component parts
5Inventory ModelsIndependent demand – finished goods, items that are ready to be soldE.g. a computerDependent demand – components of finished productsE.g. parts that make up the computer
6Types of Inventories (1 of 2) Raw materials & purchased partsPartially completed goods called work in progressFinished-goods inventories (manufacturing firms) or merchandise (retail stores)
7Types of Inventories (2 of 2) Replacement parts, tools, & suppliesGoods-in-transit to warehouses or customers
9The Material Flow Cycle (2 of 2) WaitTimeMoveTimeQueueTimeSetupTimeRunTimeInputOutputCycle TimeRun time: Job is at machine and being worked onSetup time: Job is at the work station, and the work station is being "setup."Queue time: Job is where it should be, but is not being processed because other work precedes it.Move time: The time a job spends in transitWait time: When one process is finished, but the job is waiting to be moved to the next work area.
10Performance MeasuresInventory turnover (the ratio of annual cost of goods sold to average inventory investment)Days of inventory on hand (expected number of days of sales that can be supplied from existing inventory)
11Functions of Inventory (1 of 2) To “decouple” or separate various parts of the production process, ie. to maintain independence of operationsTo meet unexpected demand & to provide high levels of customer serviceTo smooth production requirements by meeting seasonal or cyclical variations in demandTo protect against stock-outs
12Functions of Inventory (2 of 2) 5. To provide a safeguard for variation in raw material delivery time6. To provide a stock of goods that will provide a “selection” for customers7. To take advantage of economic purchase-order size8. To take advantage of quantity discounts9. To hedge against price increases
13Disadvantages of Inventory Higher costsItem cost (if purchased)Holding (or carrying) costDifficult to controlHides production problemsMay decrease flexibility
14Inventory Costs Holding (or carrying) costs Costs for storage, handling, insurance, etcSetup (or production change) costsCosts to prepare a machine or process for manufacturing an order, eg. arranging specific equipment setups, etcOrdering costs (costs of replenishing inventory)Costs of placing an order and receiving goodsShortage costsCosts incurred when demand exceeds supply5
19Shortage CostsBackordering costCost of lost sales
20Inventory Control System Defined An inventory system is the set of policies and controls that monitor levels of inventory and determine what levels should be maintained, when stock should be replenished and how large orders should beAnswers questions as:When to order?How much to order?
21Objective of Inventory Control To achieve satisfactory levels of customer service while keeping inventory costs within reasonable boundsImprove the Level of customer serviceReduce the Costs of ordering and carrying inventory
22Requirements of an Effective Inventory Management A system to keep track of inventoryA reliable forecast of demandKnowledge of lead timesReasonable estimates ofHolding costsOrdering costsShortage costsA classification system
23Inventory Counting (Control) Systems Periodic SystemPhysical count of items made at periodic intervals; order is placed for a variable amount after fixed passage of time.Perpetual (Continuous) Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item (constant amount is ordered when inventory declines to a predetermined level)
24Inventory Models Single-Period Inventory Model One time purchasing decision (Examples: selling t-shirts at a football game, newspapers, fresh bakery products, fresh flowers)Seeks to balance the costs of inventory over stock and under stockMulti-Period Inventory ModelsFixed-Order Quantity ModelsEvent triggered (Example: running out of stock)Fixed-Time Period ModelsTime triggered (Example: Monthly sales call by sales representative)7
26Single-Period Inventory Model In a single-period model, items are received in the beginning of a period and sold during the same period. The unsold items are not carried over to the next period.The unsold items may be a total waste, or sold at a reduced price, or returned to the producer at some price less than the original purchase price.The revenue generated by the unsold items is called the salvage value.
273(Newsboy Problem)Single period model: It is used to handle ordering of perishables (fresh fruits, flowers) and other items with limited useful lives (newspapers, spare parts for specialized equipment).
28Shortage cost (Cost of Understocking) Shortage cost: generally, this cost represents unrealized profit per unit(Cu=Revenue per unit – Cost per unit)If a shortage or stockout cost relates to a spare part for a machine, then shortage cost refers to the actual cost of lost production.
29Excess cost (Cost of Over Stocking) Excess cost (Ce): difference between purchase cost and salvage value of items left over at the end of a period.If there is a cost associated with disposing of excess items, the salvage cost will be negative.
30Single Period ModelGiven the costs of overestimating/underestimating demand and the probabilities of various demand sizes the goal is to identify the order quantity or stocking level that will minimize the long-run excess (overstock)or shortage costs (understock).
31Single-Period Models (Demand Distribution) Demand may be discrete or continuous. The demand of computer, newspaper, etc. is usually an integer. Such a demand is discrete. On the other hand, the demand of gasoline is not restricted to integers. Such a demand is continuous. Often, the demand of perishable food items such as fish or meat may also be continuous.Consider an order quantity QLet p = probability (demand<Q)= probability of not selling the Qth item.So, (1-p) = probability of selling the Qth item.
32Single-Period Models (Discrete Demand) Expected loss from the Qth item =Expected profit from the Qth item =So, the Qth item should be ordered ifDecision Rule (Discrete Demand):Order maximum quantity Q such thatwhere p = probability (demand<Q)
33Single-Period ModelThe service level is the probability that demand will not exceed the stocking level. The service level determines the amount of stocking level to keep.8
34Optimal Stocking Level (Choosing optimum Stocking level to minimize these costs is similar to balancing a seesaw)Service level =CsCs + CeCs = Shortage cost per unit Ce = Excess cost per unitService LevelSoQuantityCeCsBalance point
35Service Level Another way to define ‘Service Level’ is: proportion of cycles in which no stock-out occurs
36Service Level Order Cycle Demand Stock-Outs TotalSince there are two cycles out of ten in which a stockout occurs, service level is 80%. This translates to a 96% fill rate. There are a total of 1,450 units demand and 55 stockouts (which means that 1,395 units of demand are satisfied).
37Single Period Model (Demand is represented by a discrete distribution) Unlike the continuous case where the optimal solution is found by determining So which makes the distribution function equal to the critical ratio cs / (cs + ce), in the discrete case, the critical ratio takes place between two values of F(So) or F(Q)The optimal So or Q corresponds to the higher value of F( So) or F(Q).(Note that, in the discrete case, the distribution function increases by jumps)SEE EXAMPLES 17 & 18 on page 576
38Single-Period Models (Discrete Demand) Example : Demand for cookies: Demand Probability of Demand1,800 dozen 0.052,2,2,2,2,3, ,05Selling price=$0.69, cost=$0.49, salvage value=$0.29What is the optimal number of cookies to make?c
39Single-Period Models (Discrete Demand) Cs= =$0.2, Ce= =$0.2Order maximum quantity, Q such thatDemand, Q Probability(demand) Probability(demand<Q), p1,800 dozen2,2,2,2,2,3, ,
40Single-Period Models (Continous Demand) Often the demand is continuous. Even when the demand is not continuous, continuous distribution may be used because the discrete distribution may be inconvenient.We shall discuss two distributions:Uniform distributionNormal distribution
41Single-Period Models (Continuous Demand) Example 2: The J&B Card Shop sells calendars. The once-a-year order for each year’s calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated bya. uniform distribution between 150 and 850
43Single-Period Models (Continuous Demand) Now, find the Q so that p = probability(demand<Q) =0.75Q* = a+p(b-a) = ( )=675Single-Period Models (Continuous Demand)ProbabilityAreaArea ==150Demand850Q*
44Single-Period Models (Continuous Demand) Example 3: The J&B Card Shop sells calendars. The once-a-year order for each year’s calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated byb. normal distribution with = 500 and =120.
45Single-Period Models (Continuous Demand) Solution to Example 3: ce =$0.50, cu =$1.50 (see Example 2)p = = 0.75
46Single-Period Models (Continuous Demand) Now, find the Q so that p = 0.75
47Single-Period Models (Continuous Demand Q= So = mean + zσ= (120)= 582
48Single Period Example 15 (pg. 574) Demand is uniformly distributed Ce = $0.20 per unitCs = $0.60 per unitService level = Cs/(Cs+Ce) = .6/(.6+.2)Service level = .75Opt. Stock.Level=S0= ( )= 450 litersService Level = 75%QuantityCeCsStockout risk = 1.00 – 0.75 = 0.25
49Uniform Distribution [Continuous Dist’n] A random variable X is uniformly distributed on the interval (a,b), U(a,b), if its pdf and cdf are:PropertiesP(x1 < X < x2) is proportional to the length of the interval [F(x2) – F(x1) = (x2-x1)/(b-a)]E(X) = (a+b)/2 V(X) = (b-a)2/12U(0,1) provides the means to generate random numbers, from which random variates can be generated.
50Poisson Distribution [Discrete Dist’n] Poisson distribution describes many random processes quite well and is mathematically quite simple.where a > 0, pdf and cdf are:E(X) = a = V(X)
51Normal Distribution [Continuous Dist’n] A normally distributed random variable X has the pdf:Mean:Variance:Denoted as X ~ N(m,s2)Special properties:symmetric about m.The maximum value of the pdf occurs at x = m; the mean and mode are equal.
52Normal Distribution [Continuous Dist’n] Evaluating the distribution:Use numerical methods (no closed form)Independent of m and s, using the standard normal distribution:Z ~ N(0,1)Transformation of variables: let Z = (X - m) / s,
53Normal Distribution [Continuous Dist’n] Example: The time required to load an oceangoing vessel, X, is distributed as N(12,4)The probability that the vessel is loaded in less than 10 hours:Using the symmetry property, F(1) is the complement of F (-1)
54therefore we need 2,400 + .432(350) = 2,551 shirts Single Period Model (Demand is represented by a continous distribution)Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350 and we can assume that demand for shirts is approximately normally distributed. We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. What is the optimal stocking level for shirts?So =mean + zσCs = $10 and Ce = $5; P ≤ $10 / ($10 + $5) = .667Z.667 = .432therefore we need 2, (350) = 2,551 shirts
55Multi-Period Inventory Models Fixed-Order Quantity Models (Types of)Economic Order Quantity Model (EOQ)Economic Production Order Quantity (Economic Lot Size) Model (EPQ)Economic Order Quantity Model with Quantity DiscountsFixed Time Period (Fixed Order Interval) Models
56Fixed Order Quantity Models: Economic Order Quantity Model
57Economic Order Quantity Model Assumptions (1 of 2): Demand for the product is known with certainty, it is constant and uniform throughout the periodLead time (time from ordering to receipt) is known and constantPrice per unit of product is constant (no quantity discounts). So it is not included in the total cost.Inventory holding cost is based on average inventory
58Economic Order Quantity Model Assumptions (2 of 2): Ordering or setup costs are constantAll demands for the product will be satisfied (no backorders are allowed)No stockouts (shortages) are allowedThe order quantity is received all at once. (Instantaneous receipt of material in a single lot)The goal is to calculate the order quantitiy that minimizes total cost9
59Basic Fixed-Order Quantity Model and Reorder Point Behavior 4. The cycle then repeats.1. You receive an order quantity Q.R = Reorder pointQ = Economic order quantityL = Lead timeLQRTimeNumberof unitson hand(Inv.Level)2. You start using them up over time.3. When you reach down to a level of inventory of R, you place your next Q sized order.
60Average Inventory (Q/2) EOQ ModelReorder Point (ROP)TimeInventory LevelAverage Inventory (Q/2)Lead TimeOrder Quantity (Q)Demand rateOrder placedOrder received
61EOQ Cost Model: How Much to Order? By adding the holding and ordering costs together, we determine the total cost curve, which in turn is used to find the optimal order quantity that minimizes total costsSlope = 0Total CostOrder Quantity, QAnnual cost ($)Minimum total costOptimal orderQoptCarrying Cost =HQ2Ordering Cost =SDQ
62Why Holding Costs Increase? More units must be stored if more are orderedPurchase OrderDescriptionQty.Microwave1000Order quantityPurchase OrderDescriptionQty.Microwave1Order quantity
63Why Ordering Costs Decrease ? Cost is spread over more unitsExample: You need 1000 microwave ovensPurchase OrderDescriptionQty.Microwave11000 Order (Postage $ 0.33)1 Order (Postage $330)Order quantity1000
64Basic Fixed-Order Quantity (EOQ) Model Formula TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventoryTotalAnnual =CostAnnualPurchaseCostAnnualOrderingCostAnnualHoldingCost++12
65EOQ Cost ModelUsing calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of QoptTC =S DQH Q2=Q2HTCQ0 =Qopt =2SDDeriving QoptAnnual ordering cost =S DQAnnual carrying cost =HQ2Total cost =H QProving equality of costs at optimal point=S DQH Q2Q2 =2S DHQopt =2 S D
66Deriving the EOQ 1) How much to order? 2) When to order? We also need a reorder point to tell us when to place an order13
67EOQ Model Equations = × Q* D S H N T d ROP L 2 Optimal Order Quantity Expected Number of OrdersExpected Time Between OrdersWorking Days/Year=×Q*DSHNTdROPL2
68EOQ Example 1 (1 of 3)Given the information below, what are the EOQ and reorder point?Annual Demand = 1,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = $2.50Lead time = 7 daysCost per unit = $1514
69EOQ Example 1(2 of 3)In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, place the next order of 90 units when you only have 20 units left.15
70EOQ Example I(3 of 3) Order cycle time= 365/(D/Qopt) = 365/11 Orders per year = D/Qopt= 1000/90= 11 orders/yearOrder cycle time= 365/(D/Qopt) = 365/11= 33.1daysTCmin =SDQHQ2(10)(1,000)90(2,5)(90)TCmin = $ $111 = 22 $++
71EOQ Example 2(1 of 2) Annual Demand = 10,000 units Determine the economic order quantityand the reorder point given the following…Annual Demand = 10,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = 10% of cost per unitLead time = 10 daysCost per unit = $1516
72EOQ Example 2(2 of 2)Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.17
73EOQ Example 3 H = $0.75 per yard S = $150 D = 10,000 yards Qopt = 2(150)(10,000)(0.75)Qopt = 2,000 yardsTCmin =S DQH Q2TCmin =(150)(10,000)2,000(0.75)(2,000)TCmin = $750 + $750 = $1,500Orders per year = D/Qopt= 10,000/2,000= 5 orders/yearOrder cycle time =311 days/(D/Qopt)= 311/5= 62.2 store days
74When to Reorder with EOQ Ordering ? Reorder Point – is the level of inventory at which a new order is placedROP = d . LSafety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time.Service Level - Probability that demand will not exceed supply during lead time (probability that inventory available during the lead time will meet the demand) 1 - Probability of stockout
75Reorder Point Example Demand = 10,000 yards/year Store open 311 days/yearDaily demand = 10,000 / 311 = yards/dayLead time = L = 10 daysR = dL = (32.154)(10) = yards
76Determinants of the Reorder Point The rate of demandThe lead timeDemand and/or lead time variabilityStockout risk (safety stock)
77Probabilistic Models Answer how much & when to order Allow demand and lead time to varyFollows normal distributionOther EOQ assumptions applyConsider service level & safety stockService level = 1 - Probability of stockoutHigher service level means more safety stockMore safety stock means higher ROP
78Safety Stock Quantity Maximum probable demand Expected demand LTTimeExpected demandduring lead timeMaximum probable demandROPQuantitySafety stockSafety stock reduces risk ofstockout during lead time
79Reorder Point With Variable Demand point, RQLTTimeInventory level
80Reorder Point with a Safety Stock point, RQLTTimeInventory levelSafety Stock
81Reorder Point With Variable Demand and Constant Lead Time R = dL + zd Lwhered = average daily demandL = lead timed = the standard deviation of daily demandz = number of standard deviationscorresponding to the service levelprobabilityzd L = safety stock
82Reorder Point for Service Level Probability ofmeeting demand duringlead time = service levela stockoutRSafety stockdLExpected Demandzd LThe reorder point based on a normal distribution of LT demand
83Reorder Point for Variable Demand (Example) The carpet store wants a reorder point with a 95% service level and a 5% stockout probabilityd = 30 yards per day, (demand is normally distributed)d = 5 yards per dayL= 10 daysFor a 95% service level, z = 1.65R = dL + z d L= 30(10) + (1.65)(5)( 10)= yardsSafety stock = z d L= (1.65)(5)( 10)= 26.1 yards
84Shortages and Service Levels It is also important to specify: 1) Expected number of units short per order cycle E(n) =E(z) σdLT where E(z) is standardized number of units short obtained from Table 12.3, pg ) Expected number of units short per year E(N) =E (n) (D/Q) 3) Annual Service Level SLannual = 1- E(N)/D that is percentage of demand filled directly from inventory, known also as FILL RATE.
85Example 10 (pg. 568)– shortages and service levels Suppose standard deviation of lead time demand is known to be 20 units. Lead time demand is approximately normal.(a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle.(b) What lead time service level would imply an expected shortage of 2 units?
86Answer – shortage and service levels (a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle.σdLT= 20 unitslead time service level is 0.90 from z table (lead time), E(z)= (page 569, table 12.3)E(n) =E(z) σdLT = (0.048) (20)= 0.96 or about 1 unit.(b) What lead time service level would an expected shortage of 2 units imply?E(n) = 2E(n) =E(z) σdLT or E(z) = E(n) / σdLT =(2)/(20)= from the table, lead time service level is percent or 81.7%
87Shortages and Service Levels Expected number of units short per yearSee example 11, page 568Annual Service LevelSee example 12, page 570Note that annual service level will usually be grater than the cycle service level
88Fixed Order Quantity Models: -Noninstantaneous Receipt- Production Order Quantity (Economic Lot Size) Model
89Production Order Quantity Model Production done in batches or lotsCapacity to produce a part exceeds that part’s usage or demand rateAllows partial receipt of materialOther EOQ assumptions applySuited for production environmentMaterial produced, used immediatelyProvides production lot sizeLower holding cost than EOQ modelAnswers how much to order and when to order
90POQ Model Inventory Levels (1 of 2) TimeSupply BeginsSupply EndsProduction portion of cycleDemand portion of cycle with no supplyMaximum inventory level
91POQ Model Inventory Levels (2 of 2) TimeInventory LevelProduction Portion of CycleMax. Inventory Q/p·(p- u)Q*Supply BeginsSupply EndsInventory level with no demandDemand portion of cycle with no supplyAverage inventory Q/2(1- u/p)
92Maximum inventory level POQ Model Equations()-upMaximum inventory level*=1QD = Demand per yearS = Setup costH = Holding costd = Demand per dayp = Production per dayD=*SSetup CostQ()u=Holding Cost1/2 * H * Q1-p
93Production Order Quantity Example (1 of 2) H = $0.75 per yard S = $150 D = 10,000 yardsu = 10,000/311 = 32.2 yards per day p = 150 yards per dayPOQopt = = = 2,256.8 yards2 S DH 1 -up2(150)(10,000)32.2150TC = = $1,329upS DQH Q2Production run = = = days per orderQp2,256.8150
94Production Quantity Example (2 of 2) H = $0.75 per yard S = $150 D = 10,000 yardsu= 10,000/311 = 32.2 yards per day p = 150 yards per dayNumber of production runs = = = 4.43 runs/yearDQ10,0002,256.8Maximum inventory level = Q = 2,= 1,772 yardsup32.2150Qopt = = = 2,256.8 yards2CoDCc 1 -dp2(150)(10,000)32.2150TC = = $1,329CoDQCcQ2Production run = = = days per order2,256.8
95Fixed-Order Quantity Models: Economic Order Quantity Model with Quantity Discounts
96Quantity Discount Model Answers how much to order & when to orderAllows quantity discountsPrice per unit decreases as order quantity increasesOther EOQ assumptions applyTrade-off is between lower price & increased holding costTotal cost with purchasing costTC = PDS DQiP Q2Where P: Unit Price
97Total Costs with Purchasing Cost EOQTC with PDTC without PDPDQuantityAdding Purchasing cost doesn’t change EOQ
98Quantity Discount Models There are two general cases of quantity discount models:Carrying costs are constant (e.g. $2 per unit).Carrying costs are stated as a percentage off purchase price (20% of unit price)
991) Total Cost with Constant Carrying Costs (Compute the Common Optimal Order QuantityOCEOQQuantityTotal CostTCaTCcTCbDecreasingPriceCC a,b,c
1002) Total Cost with Variable Carrying Cost (Compute Optimal Order Quantity for each price range) Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula:i = percentage of unit cost attributed to carrying inventoryC = cost per unitSince “C” changes for each price-break, the formula above will have to be used with each price-break cost value
102Price-Break Example 1 (1 of 3) ORDER SIZE PRICE$10(d1)(d2)For this problem holding cost is given as a constant value, not as a percentage of price, so the optimal order quantity is the same for each of the price ranges. (see the figure 12.9)
106Price-Break Example 3 (1 of 4) A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an ordering cost of $4, a carrying cost with a rate of 2% of the unit price, and an annual demand of 10,000 units?Order Quantity(units) Price/unit($)0 to 2,499 $1.202,500 to 3,4,000 or more
107Price-Break Example (2 of 4) First, plug data into formula for each price-break value of “C”Annual Demand (D)= 10,000 unitsCost to place an order (S)= $4Carrying cost % of total cost (i)= 2%Cost per unit (C) = $1.20, $1.00, $0.98Next, determine if the computed Qopt values are feasible or notInterval from 4000 & more, the Qopt value is not feasibleInterval from , the Qopt value is not feasibleInterval from 0 to 2499, the Qopt value is feasible
108Price-Break Example 2 (3 of 4) Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?Because the total annual cost function is a “u” shaped functionTotal annual costsSo the candidates for the price-breaks are 1826, 2500, and 4000 unitsOrder Quantity
109Price-Break Example 2 (4 of 4) Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-breakTC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)= $12,043.82TC( )= $10,041TC(4000&more)= $9,949.20Finally, we select the least costly Qopt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units
110Multi-period Inventory Models: Fixed Time Period (Fixed-Order- Interval) Models
111Fixed-Order-Interval Model Orders are placed at fixed time intervalsOrder quantity for next interval? (inventory is brought up to target amount, amount ordered varies)Suppliers might encourage fixed intervalsRequires only periodic checks of inventory levels (no continous monitoring is required)Risk of stockout between intervals
112Inventory Level in a Fixed Period System Various amounts (Qi) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to target maximumpQ1Q2Q3Q4Target maximumTimed Inventory
113Fixed-Interval Benefits Tight control of inventory itemsItems from same supplier may yield savings in:OrderingPackingShipping costsMay be practical when inventories cannot be closely monitored
114Fixed-Interval Disadvantages Requires a larger safety stockIncreases carrying costCosts of periodic reviews
115Fixed-Time Period Model with Safety Stock Formula q = Average demand + Safety stock – Inventory currently on hand
116Fixed-Time Period Model: Determining the Value of sT+L The standard deviation of a sequence of random events equals the square root of the sum of the variances20
117Order Quantity for a Periodic Inventory System Q = d(tb + L) + zd T + L - Iwhered = average demand rateT = the fixed time between ordersL = lead time d = standard deviation of demandzd T + L = safety stockI = inventory levelz = the number of standard deviationsfor a specified service level
118Fixed-Period Model with Variable Demand (Example 1) d = 6 bottles per daysd = 1.2 bottlesT = 60 daysL = 5 daysI = 8 bottlesz = 1.65 (for a 95% service level)Q = d(T + L) + zd T + L - I= (6)(60 + 5) + (1.65)(1.2)= bottles
119Fixed-Time Period Model with Variable Demand (Example 2)(1 of 3) Given the information below, how many units should be ordered?Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The standard deviation of daily demand is 4 units.21
120Fixed-Time Period Model with Variable Demand (Example 2)(2 of 3) So, by looking at the value from the Table, we have a probability of , which is given by a z = 1.7522
121Fixed-Time Period Model with Variable Demand (Example 2) (3 of 3) So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period23
122ABC Classification System Demand volume and value of items varyItems kept in inventory are not of equal importance in terms of:dollars investedprofit potentialsales or usage volumestock-out penalties26
123ABC Classification System Classifying inventory according to some measure of importance and allocating control efforts accordingly.A - very importantB - mod. importantC - least importantAnnual$ valueof itemsABCHighLowPercentage of Items
124ABC AnalysisClassify inventory into 3 categories typically on the basis of the dollar value to the firm $ volume = Annual demand x Unit costA class, B class, C class Policies based on ABC analysisDevelop class A suppliers more carefullyGive tighter physical control of A itemsForecast A items more carefully
125Classifying Items as ABC 2040608010050% Annual $ UsageABCClass% $ Vol% Items70-805-1515305-1050-60% of Inventory Items
126ABC Classification PART UNIT COST ANNUAL USAGE 1 $ 60 90 2 350 40 1 $ 60 90PART UNIT COST ANNUAL USAGE
127ABC Classification PART UNIT COST ANNUAL USAGE 1 $ 60 90 2 350 40 1 $ 60 90PART UNIT COST ANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE9 $30,8 16,2 14,1 5,4 4,3 3,6 3,5 3,10 2,7 1,$85,400
128ABC Classification A B C PART UNIT COST ANNUAL USAGE 1 $ 60 90 1 $ 60 90PART UNIT COST ANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE9 $30,8 16,2 14,1 5,4 4,3 3,6 3,5 3,10 2,7 1,$85,400ABC
129ABC Classification A B C PART UNIT COST ANNUAL USAGE 1 $ 60 90 1 $ 60 90PART UNIT COST ANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE9 $30,8 16,2 14,1 5,4 4,3 3,6 3,5 3,10 2,7 1,$85,400ABC% OF TOTAL % OF TOTALCLASS ITEMS VALUE QUANTITYA 9, 8,B 1, 4,C 6, 5, 10,
130ABC Classification C B A % of Value | | | | | | 0 20 40 60 80 100 100 –80 –60 –40 –20 –0 –| | | | | |% of Quantity% of ValueABC
131Inventory Accuracy and Cycle Counting Inventory accuracy refers to how well the inventory records agree with physical count.Cycle counting refers to Physical Count of items in inventory.Used often with ABC classificationWhile A items are counted most often (e.g., daily), C items are counted the least frequently.
132Last Words Inventories have certain functions. But too much inventory Tends to hide problemsCostly to maintainSo it is desiredReduce lot sizesReduce safety stocks